Zeros Of Real Symmetric Polynomials ∗

Zeros Of Real Symmetric Polynomials ^∗

Alessandro Conflitti

Received 14 October 2005

Abstract

We consider a generalization of the symmetric polynomials and we give a sufficient condition in order to have that their real zero set contains a vector subspace of a certain dimension, which generalizes a result of R. Aron, R. Gonzalo and A. Zagorodnyuk. Furthermore, we investigate an application to elementary symmetric polynomials.

1 Introduction

The purpose of this paper is to study the (real) zero set of polynomials in nvariables invariant under the action of a given subgroup ofSn. Although this is quite a natural question (at least for symmetric polynomials, namely polynomials innvariables invari- ant under the action of the whole groupSn), it does not seem to have been studied very much. In particular, we treat on this paper the following special case of the problem:

given a certain subgroup Gof S_n and a polynomial F in n variables invariant under the action of G, determine when F⁻¹(0) contains a vector subspace. The only known result about this problem is given in [1] for a special class of symmetric polynomials:

ifF∈R[X₁, . . . , X_n] is a symmetric homogeneous polynomial having odd degree then F⁻¹(0) contains an ⁿ₂ —dimensional real vector subspace.

Furthermore we investigate an application of this problem to the elementary sym- metric polynomials: we completely describe the real zero set of e2(X1, . . . , Xn) =

1≤j<k≤nXjXk for anyn, and we present some conjectures and open questions.

Throughout the paper, Kdenotes a field of characteristic zero, except where oth- erwise stated, that is in Theorems 5 and 6, in Corollary 7, and in Conjecture 8, where K⊂R.

We note that the latter is the most interesting case, because in an algebrically closedfield these problems sometimes become trivial using tools from classical algebraic geometry.

Following [4, Chap. 1], we recall the definitions of some classical families of sym- metric polynomials.

For anyr≥1, ther—thelementary symmetric polynomial e_ris er(X1, . . . , Xn) =

1≤j1<j2<...<jr≤n

Xj1Xj2· · ·Xjr

∗Mathematics Subject Classifications: 05E05, 12D10.

†Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstraße 15, A-1090 Wien, Austria.

219

(forr= 0, we definee0= 1). Soer(X1, . . . , Xn) = 0 ifr > n.

For anyr≥1, ther—thcomplete symmetric polynomial h_r(X₁, . . . , X_n) is the sum of all monomials of degree rinX₁, . . . , X_n (forr= 0, we defineh₀= 1).

2 Main Results

Our main result holds for a class of polynomials which is more general than symmetric polynomials, namely the polynomials invariant under the following subgroup ofS_n.

DEFINITION 1. LetG^k = (1,2),(3,4), . . . ,(2k−1,2k) ≤Sn and suppose that it acts on K[X₁, . . . , X_n], with 1≤k ≤ ⁿ₂ , switching the variables X_2j−1, X_2j, for allj= 1, . . . , k.

Note that every symmetric polynomial withnvariables is invariant under the action ofGk for any 1≤k≤ ⁿ₂ .

THEOREM 2. LetF ∈K[X₁, . . . , X_n] be a polynomial invariant under the action ofGk for some 1≤k≤ ⁿ₂ , and such that writing

F(X₁, . . . , X_n) =

h1,... ,h2k≥0

c_{h1,... ,h2k}(X_2k+1, . . . , X_n)X₁^h1· · ·X_2k^h2k we have that if

h2j−1≡h2j (mod 2) f or all j= 1, . . . , k then ch1,... ,h2k(0, . . . ,0) = 0.

Then

F(α1,−α1, . . . ,αk,−αk,0, . . . ,0) = 0

for allαj∈K j = 1, . . . , k, which means thatF⁻¹(0) contains a vector subspace of dimensionk.

PROOF. We putWr={X2r+1, . . . , Xn}. Write F(X₁, . . . , X_n) =F(W₀) =

h1,h2≥0

c_h1,h2(W₁)X₁^h1X₂^h2=f₁(W₀) +F₂(W₀) where

f1(W0) =

h1,h2≥0 h1≡h2 (mod 2)

ch1,h2(W1)X₁^h1X₂^h2 (1)

and

F₂(W₀) =

h1,h2≥0 h1≡h2 (mod 2)

c_h1,h2(W₁)X₁^h1X₂^h2.

F(W0) is a polynomial invariant under the action ofG^k, so for anyα2∈K F(α2,−α2, W1) =F(−α2,α2, W1).

But the exponents ofX1, X2 inF2 are equivalent (mod 2), so it is also true that F2(α2,−α2, W1) =F2(−α2,α2, W1)

hence

f1(α2,−α2, W1) =f1(−α2,α2, W1). (2) From the definition (1), we have that the exponents of X₁ and X₂ in f₁ are not congruent (mod 2), thus

f1(α2,−α2, W1) =−f1(−α2,α2, W1). This equation and equation (2) immediately imply

f₁(α₂,−α₂, W₁) = 0 ∀α₂∈K. Now iterate the same method onF2.

On stepr≤k≤ ⁿ2 we have F(W0) =

r−1

t=1

ft(W0) +fr(W0) +Fr+1(W0) where

fr(W0) =

h1,... ,h2r≥0 h2r−1≡h2r (mod 2) h2j−1≡h2j (mod 2)

1≤j≤r−1

ch1,... ,h2r(Wr)X₁^h1· · ·X_2r^h2r, (3)

Fr+1(W0) =

h1,... ,h2r≥0 h_2j−1≡h2j (mod 2)

1≤j≤r

ch1,... ,h2r(Wr)X₁^h1· · ·X_2r^h2r

and fort= 1, . . . , r−1

f_t(α₂,−α₂, . . . ,α_2t,−α_2t, W_t) = 0 ∀α_2j∈K j= 1, . . . , t.

As before,F(W0) is a polynomial invariant under the action ofG^k so F(α₂,−α₂, . . . ,α_2r,−α_2r, W_r) =F(−α₂,α₂, . . . ,−α_2r,α_2r, W_r) for any choice ofα2j ∈K j= 1, . . . , r−1.

But for these values ft= 0 t= 1, . . . , r−1 and studying the exponents inFr+1 it is immediate that

F_r+1(α₂,−α₂, . . . ,α_2r,−α_2r, W_r) =F_r+1(−α₂,α₂, . . . ,−α_2r,α_2r, W_r), hence

fr(α2,−α2, . . . ,α2r,−α2r, Wr) =fr(−α2,α2, . . . ,−α2r,α2r, Wr). (4)

From the definition (3), we have that infrthe exponents ofX2r−1andX2rare not congruent (mod 2), whereas the exponents of X2j−1 andX2j for all 1≤j ≤r−1, have always the same parity, therefore

f_r(α₂,−α₂, . . . ,α_2r,−α_2r, W_r) =−f_r(−α₂,α₂, . . . ,−α_2r,α_2r, W_r). This equation and equation (4) immediately imply

fr(α2,−α2, . . . ,α2r,−α2r, Wr) = 0 ∀α2j ∈K j= 1, . . . , r.

So we have proved that for any k≤ ⁿ2 it is possible to write F(X1, . . . , Xn) =

k

t=1

ft(X1, . . . , Xn) +Fk+1(X1, . . . , Xn) where

Fk+1(X1, . . . , Xn) =

h1,... ,h2k≥0 h2j−1≡h2j (mod 2)

1≤j≤k

ch1,... ,h2k(Wk)X₁^h1· · ·X_2k^h2k

and fort= 1, . . . , k

f_t(α₂,−α₂, . . . ,α_2t,−α_2t, W_t) = 0 ∀α_2j∈K j= 1, . . . , t.

Ifkis as in the statement of the theorem and we putW_k = 0 (in other wordsX_2k+1= . . . = X_n = 0), we have that every c_{h1,... ,h2k}(W_k) in F_k+1 is zero because each has constant term equal to zero.

Therefore

F(α2,−α2, . . . ,α2k,−α2k,0, . . . ,0) = 0 ∀α2j∈K j= 1, . . . , k.

Theorem 2 considers a generic polynomialF ∈K[X₁, . . . , X_n] invariant under the action of Gk for some 1≤k ≤ ⁿ₂ , which avoids monomials with even degree of the form

cX₁^h1· · ·X_2k^h2k withc∈Kand h_2j−1≡h_2j (mod 2) j= 1, . . . , k. (5) Note that it is impossible to replace this hypothesis with the weaker assumption to avoid monomials of the form

c

k

j=1

(X2j−1X2j)^hj, (6)

therefore to omit the condition (mod 2), transforming the congruence in an equality.

In fact, let us consider the following example: let K ⊂ R and pr(X1, . . . , Xn) =

n

j=1X_j^rfor a generic evenr >0. Thenpris a symmetric polynomial, hence invariant

under G^k for any 1≤k ≤ ⁿ2 , it contains monomials as in (5) and no monomial as in (6), andp⁻_r¹(0) = (0, . . . ,0).

The following Corollary is immediate.

COROLLARY 3. Let F ∈ K[X₁, . . . , X_n] be a polynomial invariant under the action of G^k for some 1 ≤k ≤ ⁿ2 , and such thatF does not contain monomials of even degree. Then there is a k—dimensional vector subspace contained inF⁻¹(0).

Corollary 3 immediately implies the following result, which wasfirst proved in [1].

COROLLARY. LetF ∈ R[X₁, . . . , X_n] be a symmetric homogeneous polynomial of odd degree. Then there is an ⁿ₂ —dimensional vector subspace contained inF⁻¹(0).

The following question is quite natural:

OPEN QUESTION 4. Find other types of subgroups G < S_n such that similar results hold for polynomials invariant under their action.

We note that it seems extremely hard to have an analogue of Corollary 3 for a polynomial F ∈R[X1, . . . , Xn] having even degree, even ifF is symmetric, that is it is invariant under the action of the whole group Sn.

In fact, the following result for complete symmetric polynomials with even degree is known, see [2] for the proof.

THEOREM 5. Leth2r(X1, . . . , Xn)∈ R[X1, . . . , Xn] be the real complete sym- metric polynomial in nvariables with even degree equal to 2r.

If ⁿ_j=1X_j²= 1 then

h_2r(X₁, . . . , X_n)≥ 1 2^rr!.

Hence from the homogeneity ofh2r we have thath2r(X1, . . . , Xn)>0 if (X1, . . . , Xn) = (0, . . . ,0).

Nevertheless, we believe that it is possible tofind an analogue of Corollary 3 focused on the elementary symmetric polynomialse_2r(X₁, . . . , X_n) with even degree 2r, where on the contrary, the dimension of the maximal real vector subspace contained ine⁻_2r¹(0) depends only on the degree 2r and not on the number of variablesn.

In this sense we prove the following result.

THEOREM 6. For any n≥2, the real zero set ofe₂(X₁, . . . , X_n) is the circular cone

{X ∈Rⁿ : cos²(X, u) = 1 n}, where u= (1, . . . ,1)∈Rⁿ.

PROOF. Letnbe the number of variables ine2, , be the standard scalar product in Rⁿ, specifically v, w = ⁿ_j=1v_jw_j for any v = (v₁, . . . , v_n), w = (w₁, . . . , w_n)∈ Rⁿ, v be the euclidean norm, that is to say v = v, v , and

cos (v, w) = v, w v · w

be the cosine of the convex angle between the two vectors v, w∈Rⁿ.

We remark that 2e₂(X₁, . . . , X_n) = ⁿ_j=1X_j ²− ⁿj=1X_j², therefore X = (X1, . . . , Xn)∈e⁻₂¹(0) if and only if

⎛

⎝

n

j=1

Xj

⎞

⎠

2

=

n

j=1

X_j².

But this means that

cos²(X, u) = X, u ² X ²· u ² = 1

n, where u= (1, . . . ,1), and the desired result follows.

In particular Theorem 6 implies

COROLLARY 7. The real zero set ofe2 is a set of straight lines passing through the origin such that no three lines are on the same plane, so it contains no real vector subspace of dimension 2.

We feel that the following more general conjecture holds.

CONJECTURE 8. If r is even then e⁻_r¹(0) contains no real vector subspace of dimension r.

More precisely, symbolic computations and heuristic reasons lead us to suppose that if r≡0 (mod 2) then any vector subspace ine⁻_r¹(0) containing an element with at least rnonzero coordinates is a straight line, and obviously any vector in Rⁿ with at mostr−1 nonzero coordinates is a zero ofe_r(X₁, . . . , X_n).

We remark that the two extremal cases of Conjecture 8 are the Corollary 7 and the case e_n−1(X₁, . . . , X_n), n≡1 (mod 2) which becomes a task quite hard to tackle;

see [3] for a result aboute_n−1(X₁, . . . , X_n) with the sameflavour.

Acknowledgment. The author would like to thank Francesco Brenti for suggest- ing this problem and his helpful advice. Warm thanks are also due to the anonymous referee for his/her careful reading of the manuscript and his/her valuable suggestions.

This work is fully supported by FWF Austrian Science Fund grant P17563—N13.

References

[1] R. Aron, R. Gonzalo and A. Zagorodnyuk, Zeros of real polynomials, Linear and Multilinear Algebra, 48(2000), 107—115.

[2] D. B. Hunter, The positive—definiteness of the complete symmetric functions of even order, Math. Proc. Cambridge Philos. Soc., 82(1977), 255—258.

[3] C.-K. Li, An inequality on elementary symmetric functions, Linear and Multi- linear Algebra, 20(1987), 373—375.

[4] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford 1998.